Ring Like Ultra Large Scale Structures As Spectral Consequences Of Compact S³ Spatial Topology: A Topology Explicit Interpretation Of The Giant Arc, Big Ring, And Giant Ring

A. M. Lopez, R. G. Clowes "A Giant Ring on the sky" Recent observations at z ≈ 0. 8 reveal a striking configuration of three apparently related ultra-large-scale structures (uLSSs) within the same field: the Giant Arc (~1 Gpc), the Big Ring (~400 Mpc), and the Giant Ring (≲1 Gpc), the latter detected at >4σ statistical significance. Their nested, quasi-concentric arrangement extends well beyond the standard ΛCDM homogeneity scale (~370 Mpc) and is not reproduced as a non-random field in large-scale simulations such as FLAMINGO-10K under two-dimensional power spectrum analysis. This raises a fundamental question: are such structures anomalies of stochastic structure formation, or do they point to a deeper geometric origin? This work does not claim that these observations prove a specific global topology. Instead, it demonstrates that a compact, simply-connected spatial geometry S³ (Rₑff) provides a natural and mathematically consistent framework in which such ring-like uLSSs arise as structural features rather than statistical outliers. The key idea is that on compact manifolds, the spectrum of the Laplacian is discrete, and its eigenfunctions are globally supported. As a result, correlations are governed by spectral structure rather than by local decay mechanisms. Within the linearised free-mode approximation, four results are established: (i) Any correlation function constructed from globally supported eigenmodes on a compact connected manifold cannot exhibit a purely exponential decay of the form e^ (−r/ξ₀) at all separations. (ii) On S³ (Rₑff), the two-point correlation function is bounded, oscillatory, and does not generically decay to zero with increasing geodesic distance. (iii) A spherical shell of constant phase on S³ (Rₑff) naturally projects onto ring-like or arc-like structures when observed within a narrow redshift slice, providing a direct geometric mechanism for the emergence of observed patterns. (iv) The spectral structure admits a natural hierarchical organisation: a Fibonacci stratification of eigenmodes generates preferred geodesic scalesrₙ = πRₑff / Fₙ, with asymptotic ratio rₙ / rₙ₊₁ → φ, where φ is the golden ratio. Building on these results, a phenomenological identification is proposed in which the Giant Ring, Big Ring, and the characteristic clustering scale observed in the 2D power spectrum correspond to Fibonacci levels n = 3, 4, 5. This matching yields an effective spectral curvature scale Rₑff ≈ 475 Mpc within the z ≈ 0. 8 shell. Importantly, this effective scale is not a statement about the global curvature of the Universe and does not conflict with existing cosmological bounds such as Rglobal > 10 Gpc from Planck data. Rather, it characterises the local spectral structure relevant to the observed field. Taken together, these results suggest a shift in perspective: ultra-large-scale structures may not solely reflect the growth of local perturbations in an effectively infinite space, but can instead emerge as projections of globally organised spectral modes on a compact manifold. In this view, the observed rings are not anomalies to be explained away, but signatures of an underlying geometric order encoded in the topology and spectral properties of space itself.